Differentiating exponential and logarithmic functions · Years 12–13
In Differentiating exponential and logarithmic functions (Years 12–13) you practise 6 skills: The number e · The derivative of an exponential function · The derivative of a logarithmic function · Products and quotients · The equation of a tangent · Finding a parameter. The number e is a special number, about 2.718. The exponential function whose base is e is its own derivative: the derivative of e to the power of x is e to the power of x. The logarithm whose base is e is called the natural logarithm.
The derivative of the natural log of x is 1 over x. When the power, or the inside of the logarithm, is a function, multiply by its derivative: the derivative of e to the power of 3x is 3 times e to the power of 3x, and the derivative of the natural log of x squared plus 1 is 2x over x squared plus 1. Products, quotients and composite functions follow the same rules as for polynomials. The derivative of the product x times e to the power of x is e to the power of x plus x times e to the power of x. The derivative of the quotient the natural log of x over x is 1 minus the natural log of x, all over x squared. The derivative of the composite function e to the power of x squared is 2x times e to the power of x squared. The equation of a tangent is found as for polynomials: the gradient is the value of the derivative at the point of contact. Points where the index of e is 0, or where the inside of the logarithm is 1, keep the working simple: e to the power of 0 is 1, and the natural log of 1 is 0.
Tip: The value of e to the power of 0 is 1, and the natural log of 1 is 0. At points like these the working is easy.
What you learn
The number e
For example: Between which two whole numbers is the number e ?
- 1 < e < 2
- 3 < e < 4
- 2 < e < 3
Answer and solution
The answer: 2 < e < 3
Its value is about e ≈ 2.718 , so 2 < e < 3 .
The derivative of an exponential function
For example: Here is a function: f(x) = −4 × e^(4x) Which of these is f′(x) ?
- −4 × e^(4x)
- 4 × e^(4x)
- −16 × e^(4x − 1)
- −16 × e^(4x)
Answer and solution
The answer: −16 × e^(4x)
The derivative of the power is 4 , so multiply the function by it: (−4) × 4 × e^(4x) , that is f′(x) = −16 × e^(4x)
The derivative of a logarithmic function
For example: Here is a function: f(x) = ln(x + 6) Which of these is f′(x) ?
- x ÷ (x + 6)
- 1 ÷ x
- 1 ÷ (x + 6)²
- 1 ÷ (x + 6)
Answer and solution
The answer: 1 ÷ (x + 6)
The function inside the logarithm is x + 6 , and its derivative is 1 . So the derivative is 1 ÷ (x + 6)
Products and quotients
For example: Here is a function: f(x) = x² × ln x Which rule do you use first to differentiate it?
- The quotient rule
- The chain rule
- The sum rule
- The product rule
Answer and solution
The answer: The product rule
The function is a product of two factors, so use the product rule.
The equation of a tangent
For example: Here is a function f: 4 ln x What is the equation of the tangent to its graph at the point where x = 1 ?
Answer and solution
The answer: 4x − 4
At the point of contact, f(1) = 4 ln 1 = 0 . The derivative is 4 ÷ x , so the gradient is f′(1) = 4 . The tangent is y = 4x − 4
Finding a parameter
For example: Here is the function f(x) = a × eˣ − 5 , and f(0) = 0 . What is a ?
Answer and solution
The answer: 5
At x = 0 the value of e⁰ is 1, so a − 5 = 0 and a = 5 . Check: 5 × 1 − 5 = 0
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).